Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
798 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,7,8,6)(17,18)(19,25)(20,27)(21,29)(22,23)(24,30)(26,28)(31,32);(2,3)(6,7)(9,10)(11,15)(12,14)(13,16)(18,23)(20,25)(21,30)(26,32);(9,10,12,14)(11,13,16,15)(17,19,22,27)(18,20,23,25)(21,26,30,32)(24,28,29,31);(1,3,5,2)(4,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31);(1,4,5,8)(2,6,3,7)(9,11)(10,13)(12,16)(14,15)(17,21,22,30)(18,24,23,29)(19,26,27,32)(20,28,25,31)Polycyclic presentation:
2^8|a^2=fg,c^2=gh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#56465, #8099284, #8099326, … |
| maximal normal subgroup of | C2×#56465, #8099284, #8099326, … |
| maximal quotient of | #585580, #586295, #586775, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49453);